In a small saucepan, combine the onion, garlic, raisins and allspice. Stir in 1/2 cup water and simmer over low heat until the onion is softened and the liquid is reduced to a few tablespoons, about 15 minutes. Using a food processor, puree the mixture. Return to the saucepan; stir in the ketchup and Worcestershire sauce and simmer over low heat for 5 minutes.

Preheat the grill to medium. Rub the steaks and the mushroom caps with olive oil; season with salt and pepper. Grill the steaks, turning once, for about 15 minutes for medium-rare. Meanwhile, add the mushroom caps and grill, turning once, for 7 minutes. Slice the mushrooms; serve with the steak and steak sauce.